role of physics in science education

• Position paper
• Open access
• Published: 28 November 2019

Physics education research for 21 st century learning

• Lei Bao   ORCID: orcid.org/0000-0003-3348-4198 1 &
• Kathleen Koenig 2  

Disciplinary and Interdisciplinary Science Education Research volume  1 , Article number:  2 ( 2019 ) Cite this article

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Education goals have evolved to emphasize student acquisition of the knowledge and attributes necessary to successfully contribute to the workforce and global economy of the twenty-first Century. The new education standards emphasize higher end skills including reasoning, creativity, and open problem solving. Although there is substantial research evidence and consensus around identifying essential twenty-first Century skills, there is a lack of research that focuses on how the related subskills interact and develop over time. This paper provides a brief review of physics education research as a means for providing a context towards future work in promoting deep learning and fostering abilities in high-end reasoning. Through a synthesis of the literature around twenty-first Century skills and physics education, a set of concretely defined education and research goals are suggested for future research, along with how these may impact the next generation physics courses and how physics should be taught in the future.

Introduction

Education is the primary service offered by society to prepare its future generation workforce. The goals of education should therefore meet the demands of the changing world. The concept of learner-centered, active learning has broad, growing support in the research literature as an empirically validated teaching practice that best promotes learning for modern day students (Freeman et al., 2014 ). It stems out of the constructivist view of learning, which emphasizes that it is the learner who needs to actively construct knowledge and the teacher should assume the role of a facilitator rather than the source of knowledge. As implied by the constructivist view, learner-centered education usually emphasizes active-engagement and inquiry style teaching-learning methods, in which the learners can effectively construct their understanding under the guidance of instruction. The learner-centered education also requires educators and researchers to focus their efforts on the learners’ needs, not only to deliver effective teaching-learning approaches, but also to continuously align instructional practices to the education goals of the times. The goals of introductory college courses in science, technology, engineering, and mathematics (STEM) disciplines have constantly evolved from some notion of weed-out courses that emphasize content drilling, to the current constructivist active-engagement type of learning that promotes interest in STEM careers and fosters high-end cognitive abilities.

Following the conceptually defined framework of twenty-first Century teaching and learning, this paper aims to provide contextualized operational definitions of the goals for twenty-first Century learning in physics (and STEM in general) as well as the rationale for the importance of these outcomes for current students. Aligning to the twenty-first Century learning goals, research in physics education is briefly reviewed to provide a context towards future work in promoting deep learning and fostering abilities in high-end reasoning in parallel. Through a synthesis of the literature around twenty-first Century skills and physics education, a set of concretely defined education and research goals are suggested for future research. These goals include: domain-specific research in physics learning; fostering scientific reasoning abilities that are transferable across the STEM disciplines; and dissemination of research-validated curriculum and approaches to teaching and learning. Although this review has a focus on physics education research (PER), it is beneficial to expand the perspective to view physics education in the broader context of STEM learning. Therefore, much of the discussion will blend PER with STEM education as a continuum body of work on teaching and learning.

Education goals for twenty-first century learning

Education goals have evolved to emphasize student acquisition of essential “21 st Century skills”, which define the knowledge and attributes necessary to successfully contribute to the workforce and global economy of the 21st Century (National Research Council, 2011 , 2012a ). In general, these standards seek to transition from emphasizing content-based drilling and memorization towards fostering higher-end skills including reasoning, creativity, and open problem solving (United States Chamber of Commerce, 2017 ). Initiatives on advancing twenty-first Century education focus on skills that converge on three broad clusters: cognitive, interpersonal, and intrapersonal, all of which include a rich set of sub-dimensions.

Within the cognitive domain, multiple competencies have been proposed, including deep learning, non-routine problem solving, systems thinking, critical thinking, computational and information literacy, reasoning and argumentation, and innovation (National Research Council, 2012b ; National Science and Technology Council, 2018 ). Interpersonal skills are those necessary for relating to others, including the ability to work creatively and collaboratively as well as communicate clearly. Intrapersonal skills, on the other hand, reside within the individual and include metacognitive thinking, adaptability, and self-management. These involve the ability to adjust one’s strategy or approach along with the ability to work towards important goals without significant distraction, both essential for sustained success in long-term problem solving and career development.

Although many descriptions exist for what qualifies as twenty-first Century skills, student abilities in scientific reasoning and critical thinking are the most commonly noted and widely studied. They are highly connected with the other cognitive skills of problem solving, decision making, and creative thinking (Bailin, 1996 ; Facione, 1990 ; Fisher, 2001 ; Lipman, 2003 ; Marzano et al., 1988 ), and have been important educational goals since the 1980s (Binkley et al., 2010 ; NCET, 1987 ). As a result, they play a foundational role in defining, assessing, and developing twenty-first Century skills.

The literature for critical thinking is extensive (Bangert-Drowns & Bankert, 1990 ; Facione, 1990 ; Glaser, 1941 ). Various definitions exist with common underlying principles. Broadly defined, critical thinking is the application of the cognitive skills and strategies that aim for and support evidence-based decision making. It is the thinking involved in solving problems, formulating inferences, calculating likelihoods, and making decisions (Halpern, 1999 ). It is the “reasonable reflective thinking focused on deciding what to believe or do” (Ennis, 1993 ). Critical thinking is recognized as a way to understand and evaluate subject matter; producing reliable knowledge and improving thinking itself (Paul, 1990 ; Siegel, 1988 ).

The notion of scientific reasoning is often used to label the set of skills that support critical thinking, problem solving, and creativity in STEM. Broadly defined, scientific reasoning includes the thinking and reasoning skills involved in inquiry, experimentation, evidence evaluation, inference and argument that support the formation and modification of concepts and theories about the natural world; such as the ability to systematically explore a problem, formulate and test hypotheses, manipulate and isolate variables, and observe and evaluate consequences (Bao et al., 2009 ; Zimmerman, 2000 ). Critical thinking and scientific reasoning share many features, where both emphasize evidence-based decision making in multivariable causal conditions. Critical thinking can be promoted through the development of scientific reasoning, which includes student ability to reach a reliable conclusion after identifying a question, formulating hypotheses, gathering relevant data, and logically testing and evaluating the hypothesis. In this way, scientific reasoning can be viewed as a scientific domain instantiation of critical thinking in the context of STEM learning.

In STEM learning, cognitive aspects of the twenty-first Century skills aim to develop reasoning skills, critical thinking skills, and deep understanding, all of which allow students to develop well connected expert-like knowledge structures and engage in meaningful scientific inquiry and problem solving. Within physics education, a core component of STEM education, the learning of conceptual understanding and problem solving remains a current emphasis. However, the fast-changing work environment and technology-driven world require a new set of core knowledge, skills, and habits of mind to solve complex interdisciplinary problems, gather and evaluate evidence, and make sense of information from a variety of sources (Tanenbaum, 2016 ). The education goals in physics are transitioning towards ability fostering as well as extension and integration with other STEM disciplines. Although curriculum that supports these goals is limited, there are a number of attempts, particularly in developing active learning classrooms and inquiry-based laboratory activities, which have demonstrated success. Some of these are described later in this paper as they provide a foundation for future work in physics education.

Interpersonal skills, such as communication and collaboration, are also essential for twenty-first Century problem-solving tasks, which are often open-ended, complex, and team-based. As the world becomes more connected in a multitude of dimensions, tackling significant problems involving complex systems often goes beyond the individual and requires working with others who are increasingly from culturally diverse backgrounds. Due to the rise of communication technologies, being able to articulate thoughts and ideas in a variety of formats and contexts is crucial, as well as the ability to effectively listen or observe to decipher meaning. Interpersonal skills can be promoted by integrating group-learning experiences into the classroom setting, while providing students with the opportunity to engage in open-ended tasks with a team of peer learners who may propose more than one plausible solution. These experiences should be designed such that students must work collaboratively and responsibly in teams to develop creative solutions, which are later disseminated through informative presentations and clearly written scientific reports. Although educational settings in general have moved to providing students with more and more opportunities for collaborative learning, a lack of effective assessments for these important skills has been a limiting factor for producing informative research and widespread implementation. See Liu ( 2010 ) for an overview of measurement instruments reported in the research literature.

Intrapersonal skills are based on the individual and include the ability to manage one’s behavior and emotions to achieve goals. These are especially important for adapting in the fast-evolving collaborative modern work environment and for learning new tasks to solve increasingly challenging interdisciplinary problems, both of which require intellectual openness, work ethic, initiative, and metacognition, to name a few. These skills can be promoted using instruction which, for example, includes metacognitive learning strategies, provides opportunities to make choices and set goals for learning, and explicitly connects to everyday life events. However, like interpersonal skills, the availability of relevant assessments challenges advancement in this area. In this review, the vast amount of studies on interpersonal and intrapersonal skills will not be discussed in order to keep the main focus on the cognitive side of skills and reasoning.

The purpose behind discussing twenty-first Century skills is that this set of skills provides important guidance for establishing essential education goals for modern society and learners. However, although there is substantial research evidence and consensus around identifying necessary twenty-first Century skills, there is a lack of research that focuses on how the related subskills interact and develop over time (Reimers & Chung, 2016 ), with much of the existing research residing in academic literature that is focused on psychology rather than education systems (National Research Council, 2012a ). Therefore, a major and challenging task for discipline-based education researchers and educators is to operationally define discipline-specific goals that align with the twenty-first Century skills for each of the STEM fields. In the following sections, this paper will provide a limited vision of the research endeavors in physics education that can translate the past and current success into sustained impact for twenty-first Century teaching and learning.

Proposed education and research goals

Physics education research (PER) is often considered an early pioneer in discipline-based education research (National Research Council, 2012c ), with well-established, broad, and influential outcomes (e.g., Hake, 1998 ; Hsu, Brewe, Foster, & Harper, 2004 ; McDermott & Redish, 1999 ; Meltzer & Thornton, 2012 ). Through the integration of twenty-first Century skills with the PER literature, a set of broadly defined education and research goals is proposed for future PER work:

Discipline-specific deep learning: Cognitive and education research involving physics learning has established a rich literature on student learning behaviors along with a number of frameworks. Some of the popular frameworks include conceptual understanding and concept change, problem solving, knowledge structure, deep learning, and knowledge integration. Aligned with twenty-first Century skills, future research in physics learning should aim to integrate the multiple areas of existing work, such that they help students develop well integrated knowledge structures in order to achieve deep leaning in physics.

Fostering scientific reasoning for transfer across STEM disciplines: The broad literature in physics learning and scientific reasoning can provide a solid foundation to further develop effective physics education approaches, such as active engagement instruction and inquiry labs, specifically targeting scientific inquiry abilities and reasoning skills. Since scientific reasoning is a more domain-general cognitive ability, success in physics can also more readily inform research and education practices in other STEM fields.

Research, development, assessment, and dissemination of effective education approaches: Developing and maintaining a supportive infrastructure of education research and implementation has always been a challenge, not only in physics but in all STEM areas. The twenty-first Century education requires researchers and instructors across STEM to work together as an extended community in order to construct a sustainable integrated STEM education environment. Through this new infrastructure, effective team-based inquiry learning and meaningful assessment can be delivered to help students develop a comprehensive skills set including deep understanding and scientific reasoning, as well as communication and other non-cognitive abilities.

The suggested research will generate understanding and resources to support education practices that meet the requirements of the Next Generation Science Standards (NGSS), which explicitly emphasize three areas of learning including disciplinary core ideas, crosscutting concepts, and practices (National Research Council, 2012b ). The first goal for promoting deep learning of disciplinary knowledge corresponds well to the NGSS emphasis on disciplinary core ideas, which play a central role in helping students develop well integrated knowledge structures to achieve deep understanding. The second goal on fostering transferable scientific reasoning skills supports the NGSS emphasis on crosscutting concepts and practices. Scientific reasoning skills are crosscutting cognitive abilities that are essential to the development of domain-general concepts and modeling strategies. In addition, the development of scientific reasoning requires inquiry-based learning and practices. Therefore, research on scientific reasoning can produce a valuable knowledge base on education means that are effective for developing crosscutting concepts and promoting meaningful practices in STEM. The third research goal addresses the challenge in the assessment of high-end skills and the dissemination of effective educational approaches, which supports all NGSS initiatives to ensure sustainable development and lasting impact. The following sections will discuss the research literature that provides the foundation for these three research goals and identify the specific challenges that will need to be addressed in future work.

Promoting deep learning in physics education

Physics education for the twenty-first Century aims to foster high-end reasoning skills and promote deep conceptual understanding. However, many traditional education systems place strong emphasis on only problem solving with the expectation that students obtain deep conceptual understanding through repetitive problem-solving practices, which often doesn’t occur (Alonso, 1992 ). This focus on problem solving has been shown to have limitations as a number of studies have revealed disconnections between learning conceptual understanding and problem-solving skills (Chiu, 2001 ; Chiu, Guo, & Treagust, 2007 ; Hoellwarth, Moelter, & Knight, 2005 ; Kim & Pak, 2002 ; Nakhleh, 1993 ; Nakhleh & Mitchell, 1993 ; Nurrenbern & Pickering, 1987 ; Stamovlasis, Tsaparlis, Kamilatos, Papaoikonomou, & Zarotiadou, 2005 ). In fact, drilling in problem solving may actually promote memorization of context-specific solutions with minimal generalization rather than transitioning students from novices to experts.

Towards conceptual understanding and learning, many models and definitions have been established to study and describe student conceptual knowledge states and development. For example, students coming into a physics classroom often hold deeply rooted, stable understandings that differ from expert conceptions. These are commonly referred to as misconceptions or alternative conceptions (Clement, 1982 ; Duit & Treagust, 2003 ; Dykstra Jr, Boyle, & Monarch, 1992 ; Halloun & Hestenes, 1985a , 1985b ). Such students’ conceptions are context dependent and exist as disconnected knowledge fragments, which are strongly situated within specific contexts (Bao & Redish, 2001 , 2006 ; Minstrell, 1992 ).

In modeling students’ knowledge structures, DiSessa’s proposed phenomenological primitives (p-prim) describe a learner’s implicit thinking, cued from specific contexts, as an underpinning cognitive construct for a learner’s expressed conception (DiSessa, 1993 ; Smith III, DiSessa, & Roschelle, 1994 ). Facets, on the other hand, map between the implicit p-prim and concrete statements of beliefs and are developed as discrete and independent units of thought, knowledge, or strategies used by individuals to address specific situations (Minstrell, 1992 ). Ontological categories, defined by Chi, describe student reasoning in the most general sense. Chi believed that these are distinct, stable, and constraining, and that a core reason behind novices’ difficulties in physics is that they think of physics within the category of matter instead of processes (Chi, 1992 ; Chi & Slotta, 1993 ; Chi, Slotta, & De Leeuw, 1994 ; Slotta, Chi, & Joram, 1995 ). More details on conceptual learning and problem solving are well summarized in the literature (Hsu et al., 2004 ; McDermott & Redish, 1999 ), from which a common theme emerges from the models and definitions. That is, learning is context dependent and students with poor conceptual understanding typically have locally connected knowledge structures with isolated conceptual constructs that are unable to establish similarities and contrasts between contexts.

Additionally, this idea of fragmentation is demonstrated through many studies on student problem solving in physics and other fields. It has been shown that a student’s knowledge organization is a key aspect for distinguishing experts from novices (Bagno, Eylon, & Ganiel, 2000 ; Chi, Feltovich, & Glaser, 1981 ; De Jong & Ferguson-Hesler, 1986 ; Eylon & Reif, 1984 ; Ferguson-Hesler & De Jong, 1990 ; Heller & Reif, 1984 ; Larkin, McDermott, Simon, & Simon, 1980 ; Smith, 1992 ; Veldhuis, 1990 ; Wexler, 1982 ). Expert’s knowledge is organized around core principles of physics, which are applied to guide problem solving and develop connections between different domains as well as new, unfamiliar situations (Brown, 1989 ; Perkins & Salomon, 1989 ; Salomon & Perkins, 1989 ). Novices, on the other hand, lack a well-organized knowledge structure and often solve problems by relying on surface features that are directly mapped to certain problem-solving outcomes through memorization (Chi, Bassok, Lewis, Reimann, & Glaser, 1989 ; Hardiman, Dufresne, & Mestre, 1989 ; Schoenfeld & Herrmann, 1982 ).

This lack of organization creates many difficulties in the comprehension of basic concepts and in solving complex problems. This leads to the common complaint that students’ knowledge of physics is reduced to formulas and vague labels of the concepts, which are unable to substantively contribute to meaningful reasoning processes. A novice’s fragmented knowledge structure severely limits the learner’s conceptual understanding. In essence, these students are able to memorize how to approach a problem given specific information but lack the understanding of the underlying concept of the approach, limiting their ability to apply this approach to a novel situation. In order to achieve expert-like understanding, a student’s knowledge structure must integrate all of the fragmented ideas around the core principle to form a coherent and fully connected conceptual framework.

Towards a more general theoretical consideration, students’ alternative conceptions and fragmentation in knowledge structures can be viewed through both the “naïve theory” framework (e.g., Posner, Strike, Hewson, & Gertzog, 1982 ; Vosniadou, Vamvakoussi, & Skopeliti, 2008 ) and the “knowledge in pieces” (DiSessa, 1993 ) perspective. The “naïve theory” framework considers students entering the classroom with stable and coherent ideas (naïve theories) about the natural world that differ from those presented by experts. In the “knowledge in pieces” perspective, student knowledge is constructed in real-time and incorporates context features with the p-prims to form the observed conceptual expressions. Although there exists an ongoing debate between these two views (Kalman & Lattery, 2018 ), it is more productive to focus on their instructional implications for promoting meaningful conceptual change in students’ knowledge structures.

In the process of learning, students may enter the classroom with a range of initial states depending on the population and content. For topics with well-established empirical experiences, students often have developed their own ideas and understanding, while on topics without prior exposure, students may create their initial understanding in real-time based on related prior knowledge and given contextual features (Bao & Redish, 2006 ). These initial states of understanding, regardless of their origin, are usually different from those of experts. Therefore, the main function of teaching and learning is to guide students to modify their initial understanding towards the experts’ views. Although students’ initial understanding may exist as a body of coherent ideas within limited contexts, as students start to change their knowledge structures throughout the learning process, they may evolve into a wide range of transitional states with varying levels of knowledge integration and coherence. The discussion in this brief review on students’ knowledge structures regarding fragmentation and integration are primarily focused on the transitional stages emerged through learning.

The corresponding instructional goal is then to help students more effectively develop an integrated knowledge structure so as to achieve a deep conceptual understanding. From an educator’s perspective, Bloom’s taxonomy of education objectives establishes a hierarchy of six levels of cognitive skills based on their specificity and complexity: Remember (lowest and most specific), Understand, Apply, Analyze, Evaluate, and Create (highest and most general and complex) (Anderson et al., 2001 ; Bloom, Engelhart, Furst, Hill, & Krathwohl, 1956 ). This hierarchy of skills exemplifies the transition of a learner’s cognitive development from a fragmented and contextually situated knowledge structure (novice with low level cognitive skills) to a well-integrated and globally networked expert-like structure (with high level cognitive skills).

As a student’s learning progresses from lower to higher cognitive levels, the student’s knowledge structure becomes more integrated and is easier to transfer across contexts (less context specific). For example, beginning stage students may only be able to memorize and perform limited applications of the features of certain contexts and their conditional variations, with which the students were specifically taught. This leads to the establishment of a locally connected knowledge construct. When a student’s learning progresses from the level of Remember to Understand, the student begins to develop connections among some of the fragmented pieces to form a more fully connected network linking a larger set of contexts, thus advancing into a higher level of understanding. These connections and the ability to transfer between different situations form the basis of deep conceptual understanding. This growth of connections leads to a more complete and integrated cognitive structure, which can be mapped to a higher level on Bloom’s taxonomy. This occurs when students are able to relate a larger number of different contextual and conditional aspects of a concept for analyzing and evaluating to a wider variety of problem situations.

Promoting the growth of connections would appear to aid in student learning. Exactly which teaching methods best facilitate this are dependent on the concepts and skills being learned and should be determined through research. However, it has been well recognized that traditional instruction often fails to help students obtain expert-like conceptual understanding, with many misconceptions still existing after instruction, indicating weak integration within a student’s knowledge structure (McKeachie, 1986 ).

Recognizing the failures of traditional teaching, various research-informed teaching methods have been developed to enhance student conceptual learning along with diagnostic tests, which aim to measure the existence of misconceptions. Most advances in teaching methods focus on the inclusion of inquiry-based interactive-engagement elements in lecture, recitations, and labs. In physics education, these methods were popularized after Hake’s landmark study demonstrated the effectiveness of interactive-engagement over traditional lectures (Hake, 1998 ). Some of these methods include the use of peer instruction (Mazur, 1997 ), personal response systems (e.g., Reay, Bao, Li, Warnakulasooriya, & Baugh, 2005 ), studio-style instruction (Beichner et al., 2007 ), and inquiry-based learning (Etkina & Van Heuvelen, 2001 ; Laws, 2004 ; McDermott, 1996 ; Thornton & Sokoloff, 1998 ). The key approach of these methods aims to improve student learning by carefully targeting deficits in student knowledge and actively encouraging students to explore and discuss. Rather than rote memorization, these approaches help promote generalization and deeper conceptual understanding by building connections between knowledge elements.

Based on the literature, including Bloom’s taxonomy and the new education standards that emphasize twenty-first Century skills, a common focus on teaching and learning can be identified. This focus emphasizes helping students develop connections among fragmented segments of their knowledge pieces and is aligned with the knowledge integration perspective, which focuses on helping students develop and refine their knowledge structure toward a more coherently organized and extensively connected network of ideas (Lee, Liu, & Linn, 2011 ; Linn, 2005 ; Nordine, Krajcik, & Fortus, 2011 ; Shen, Liu, & Chang, 2017 ). For meaningful learning to occur, new concepts must be integrated into a learner’s existing knowledge structure by linking the new knowledge to already understood concepts.

Forming an integrated knowledge structure is therefore essential to achieving deep learning, not only in physics but also in all STEM fields. However, defining what connections must occur at different stages of learning, as well as understanding the instructional methods necessary for effectively developing such connections within each STEM disciplinary context, are necessary for current and future research. Together these will provide the much needed foundational knowledge base to guide the development of the next generation of curriculum and classroom environment designed around twenty-first Century learning.

Developing scientific reasoning with inquiry labs

Scientific reasoning is part of the widely emphasized cognitive strand of twenty-first Century skills. Through development of scientific reasoning skills, students’ critical thinking, open-ended problem-solving abilities, and decision-making skills can be improved. In this way, targeting scientific reasoning as a curricular objective is aligned with the goals emphasized in twenty-first Century education. Also, there is a growing body of research on the importance of student development of scientific reasoning, which have been found to positively correlate with course achievement (Cavallo, Rozman, Blickenstaff, & Walker, 2003 ; Johnson & Lawson, 1998 ), improvement on concept tests (Coletta & Phillips, 2005 ; She & Liao, 2010 ), engagement in higher levels of problem solving (Cracolice, Deming, & Ehlert, 2008 ; Fabby & Koenig, 2013 ); and success on transfer (Ates & Cataloglu, 2007 ; Jensen & Lawson, 2011 ).

Unfortunately, research has shown that college students are lacking in scientific reasoning. Lawson ( 1992 ) found that ~ 50% of intro biology students are not capable of applying scientific reasoning in learning, including the ability to develop hypotheses, control variables, and design experiments; all necessary for meaningful scientific inquiry. Research has also found that traditional courses do not significantly develop these abilities, with pre-to-post-test gains of 1%–2%, while inquiry-based courses have gains around 7% (Koenig, Schen, & Bao, 2012 ; Koenig, Schen, Edwards, & Bao, 2012 ). Others found that undergraduates have difficulty developing evidence-based decisions and differentiating between and linking evidence with claims (Kuhn, 1992 ; Shaw, 1996 ; Zeineddin & Abd-El-Khalick, 2010 ). A large scale international study suggested that learning of physics content knowledge with traditional teaching practices does not improve students’ scientific reasoning skills (Bao et al., 2009 ).

Aligned to twenty-first Century learning, it is important to implement curriculum that is specifically designed for developing scientific reasoning abilities within current education settings. Although traditional lectures may continue for decades due to infrastructure constraints, a unique opportunity can be found in the lab curriculum, which may be more readily transformed to include hands-on minds-on group learning activities that are ideal for developing students’ abilities in scientific inquiry and reasoning.

For well over a century, the laboratory has held a distinctive role in student learning (Meltzer & Otero, 2015 ). However, many existing labs, which haven’t changed much since the late 1980s, have received criticism for their outdated cookbook style that lacks effectiveness in developing high-end skills. In addition, labs have been primarily used as a means for verifying the physical principles presented in lecture, and unfortunately, Hofstein and Lunetta ( 1982 ) found in an early review of the literature that research was unable to demonstrate the impact of the lab on student content learning.

About this same time, a shift towards a constructivist view of learning gained popularity and influenced lab curriculum development towards engaging students in the process of constructing knowledge through science inquiry. Curricula, such as Physics by Inquiry (McDermott, 1996 ), Real-Time Physics (Sokoloff, Thornton, & Laws, 2011 ), and Workshop Physics (Laws, 2004 ), were developed with a primary focus on engaging students in cognitive conflict to address misconceptions. Although these approaches have been shown to be highly successful in improving deep learning of physics concepts (McDermott & Redish, 1999 ), the emphasis on conceptual learning does not sufficiently impact the domain general scientific reasoning skills necessitated in the goals of twenty-first Century learning.

Reform in science education, both in terms of targeted content and skills, along with the emergence of knowledge regarding human cognition and learning (Bransford, Brown, & Cocking, 2000 ), have generated renewed interest in the potential of inquiry-based lab settings for skill development. In these types of hands-on minds-on learning, students apply the methods and procedures of science inquiry to investigate phenomena and construct scientific claims, solve problems, and communicate outcomes, which holds promise for developing both conceptual understanding and scientific reasoning skills in parallel (Trowbridge, Bybee, & Powell, 2000 ). In addition, the availability of technology to enhance inquiry-based learning has seen exponential growth, along with the emergence of more appropriate research methodologies to support research on student learning.

Although inquiry-based labs hold promise for developing students’ high-end reasoning, analytic, and scientific inquiry abilities, these educational endeavors have not become widespread, with many existing physics laboratory courses still viewed merely as a place to illustrate the physical principles from the lecture course (Meltzer & Otero, 2015 ). Developing scientific ideas from practical experiences, however, is a complex process. Students need sufficient time and opportunity for interaction and reflection on complex, investigative tasks. Blended learning, which merges lecture and lab (such as studio style courses), addresses this issue to some extent, but has experienced limited adoption, likely due to the demanding infrastructure resources, including dedicated technology-intensive classroom space, equipment and maintenance costs, and fully committed trained staff.

Therefore, there is an immediate need to transform the existing standalone lab courses, within the constraints of the existing education infrastructure, into more inquiry-based designs, with one of its primary goals dedicated to developing scientific reasoning skills. These labs should center on constructing knowledge, along with hands-on minds-on practical skills and scientific reasoning, to support modeling a problem, designing and implementing experiments, analyzing and interpreting data, drawing and evaluating conclusions, and effective communication. In particular, training on scientific reasoning needs to be explicitly addressed in the lab curriculum, which should contain components specifically targeting a set of operationally-defined scientific reasoning skills, such as ability to control variables or engage in multivariate causal reasoning. Although effective inquiry may also implicitly develop some aspects of scientific reasoning skills, such development is far less efficient and varies with context when the primary focus is on conceptual learning.

Several recent efforts to enhance the standalone lab course have shown promise in supporting education goals that better align with twenty-first Century learning. For example, the Investigative Science Learning Environment (ISLE) labs involve a series of tasks designed to help students develop the “habits of mind” of scientists and engineers (Etkina et al., 2006 ). The curriculum targets reasoning as well as the lab learning outcomes published by the American Association of Physics Teachers (Kozminski et al., 2014 ). Operationally, ISLE methods focus on scaffolding students’ developing conceptual understanding using inquiry learning without a heavy emphasis on cognitive conflict, making it more appropriate and effective for entry level students and K-12 teachers.

Likewise, Koenig, Wood, Bortner, and Bao ( 2019 ) have developed a lab curriculum that is intentionally designed around the twenty-first Century learning goals for developing cognitive, interpersonal, and intrapersonal abilities. In terms of the cognitive domain, the lab learning outcomes center on critical thinking and scientific reasoning but do so through operationally defined sub-skills, all of which are transferrable across STEM. These selected sub-skills are found in the research literature, and include the ability to control variables and engage in data analytics and causal reasoning. For each targeted sub-skill, a series of pre-lab and in-class activities provide students with repeated, deliberate practice within multiple hypothetical science-based scenarios followed by real inquiry-based lab contexts. This explicit instructional strategy has been shown to be essential for the development of scientific reasoning (Chen & Klahr, 1999 ). In addition, the Karplus Learning Cycle (Karplus, 1964 ) provides the foundation for the structure of the lab activities and involves cycles of exploration, concept introduction, and concept application. The curricular framework is such that as the course progresses, the students engage in increasingly complex tasks, which allow students the opportunity to learn gradually through a progression from simple to complex skills.

As part of this same curriculum, students’ interpersonal skills are developed, in part, through teamwork, as students work in groups of 3 or 4 to address open-ended research questions, such as, What impacts the period of a pendulum? In addition, due to time constraints, students learn early on about the importance of working together in an efficient manor towards a common goal, with one set of written lab records per team submitted after each lab. Checkpoints built into all in-class activities involve Socratic dialogue between the instructor and students and promote oral communication. This use of directed questioning guides students in articulating their reasoning behind decisions and claims made, while supporting the development of scientific reasoning and conceptual understanding in parallel (Hake, 1992 ). Students’ intrapersonal skills, as well as communication skills, are promoted through the submission of individual lab reports. These reports require students to reflect upon their learning over each of four multi-week experiments and synthesize their ideas into evidence-based arguments, which support a claim. Due to the length of several weeks over which students collect data for each of these reports, the ability to organize the data and manage their time becomes essential.

Despite the growing emphasis on research and development of curriculum that targets twenty-first Century learning, converting a traditionally taught lab course into a meaningful inquiry-based learning environment is challenging in current reform efforts. Typically, the biggest challenge is a lack of resources; including faculty time to create or adapt inquiry-based materials for the local setting, training faculty and graduate student instructors who are likely unfamiliar with this approach, and the potential cost of new equipment. Koenig et al. ( 2019 ) addressed these potential implementation barriers by designing curriculum with these challenges in mind. That is, the curriculum was designed as a flexible set of modules that target specific sub-skills, with each module consisting of pre-lab (hypothetical) and in-lab (real) activities. Each module was designed around a curricular framework such that an adopting institution can use the materials as written, or can incorporate their existing equipment and experiments into the framework with minimal effort. Other non-traditional approaches have also been experimented with, such as the work by Sobhanzadeh, Kalman, and Thompson ( 2017 ), which targets typical misconceptions by using conceptual questions to engage students in making a prediction, designing and conducting a related experiment, and determining whether or not the results support the hypothesis.

Another challenge for inquiry labs is the assessment of skills-based learning outcomes. For assessment of scientific reasoning, a new instrument on inquiry in scientific thinking analytics and reasoning (iSTAR) has been developed, which can be easily implemented across large numbers of students as both a pre- and post-test to assess gains. iSTAR assesses reasoning skills necessary in the systematical conduct of scientific inquiry, which includes the ability to explore a problem, formulate and test hypotheses, manipulate and isolate variables, and observe and evaluate the consequences (see www.istarassessment.org ). The new instrument expands upon the commonly used classroom test of scientific reasoning (Lawson, 1978 , 2000 ), which has been identified with a number of validity weaknesses and a ceiling effect for college students (Bao, Xiao, Koenig, & Han, 2018 ).

Many education innovations need supporting infrastructures that can ensure adoption and lasting impact. However, making large-scale changes to current education settings can be risky, if not impossible. New education approaches, therefore, need to be designed to adapt to current environmental constraints. Since higher-end skills are a primary focus of twenty-first Century learning, which are most effectively developed in inquiry-based group settings, transforming current lecture and lab courses into this new format is critical. Although this transformation presents great challenges, promising solutions have already emerged from various research efforts. Perhaps the biggest challenge is for STEM educators and researchers to form an alliance to work together to re-engineer many details of the current education infrastructure in order to overcome the multitude of implementation obstacles.

This paper attempts to identify a few central ideas to provide a broad picture for future research and development in physics education, or STEM education in general, to promote twenty-first Century learning. Through a synthesis of the existing literature within the authors’ limited scope, a number of views surface.

Education is a service to prepare (not to select) the future workforce and should be designed as learner-centered, with the education goals and teaching-learning methods tailored to the needs and characteristics of the learners themselves. Given space constraints, the reader is referred to the meta-analysis conducted by Freeman et al. ( 2014 ), which provides strong support for learner-centered instruction. The changing world of the twenty-first Century informs the establishment of new education goals, which should be used to guide research and development of teaching and learning for present day students. Aligned to twenty-first Century learning, the new science standards have set the goals for STEM education to transition towards promoting deep learning of disciplinary knowledge, thereby building upon decades of research in PER, while fostering a wide range of general high-end cognitive and non-cognitive abilities that are transferable across all disciplines.

Following these education goals, more research is needed to operationally define and assess the desired high-end reasoning abilities. Building on a clear definition with effective assessments, a large number of empirical studies are needed to investigate how high-end abilities can be developed in parallel with deep learning of concepts, such that what is learned can be generalized to impact the development of curriculum and teaching methods which promote skills-based learning across all STEM fields. Specifically for PER, future research should emphasize knowledge integration to promote deep conceptual understanding in physics along with inquiry learning to foster scientific reasoning. Integration of physics learning in contexts that connect to other STEM disciplines is also an area for more research. Cross-cutting, interdisciplinary connections are becoming important features of the future generation physics curriculum and defines how physics should be taught collaboratively with other STEM courses.

This paper proposed meaningful areas for future research that are aligned with clearly defined education goals for twenty-first Century learning. Based on the existing literature, a number of challenges are noted for future directions of research, including the need for:

clear and operational definitions of goals to guide research and practice

concrete operational definitions of high-end abilities for which students are expected to develop

effective assessment methods and instruments to measure high-end abilities and other components of twenty-first Century learning

a knowledge base of the curriculum and teaching and learning environments that effectively support the development of advanced skills

integration of knowledge and ability development regarding within-discipline and cross-discipline learning in STEM

effective means to disseminate successful education practices

The list is by no means exhaustive, but these themes emerge above others. In addition, the high-end abilities discussed in this paper focus primarily on scientific reasoning, which is highly connected to other skills, such as critical thinking, systems thinking, multivariable modeling, computational thinking, design thinking, etc. These abilities are expected to develop in STEM learning, although some may be emphasized more within certain disciplines than others. Due to the limited scope of this paper, not all of these abilities were discussed in detail but should be considered an integral part of STEM learning.

Finally, a metacognitive position on education research is worth reflection. One important understanding is that the fundamental learning mechanism hasn’t changed, although the context in which learning occurs has evolved rapidly as a manifestation of the fast-forwarding technology world. Since learning is a process at the interface between a learner’s mind and the environment, the main focus of educators should always be on the learner’s interaction with the environment, not just the environment. In recent education developments, many new learning platforms have emerged at an exponential rate, such as the massive open online courses (MOOCs), STEM creative labs, and other online learning resources, to name a few. As attractive as these may be, it is risky to indiscriminately follow trends in education technology and commercially-incentivized initiatives before such interventions are shown to be effective by research. Trends come and go but educators foster students who have only a limited time to experience education. Therefore, delivering effective education is a high-stakes task and needs to be carefully and ethically planned and implemented. When game-changing opportunities emerge, one needs to not only consider the winners (and what they can win), but also the impact on all that is involved.

Based on a century of education research, consensus has settled on a fundamental mechanism of teaching and learning, which suggests that knowledge is developed within a learner through constructive processes and that team-based guided scientific inquiry is an effective method for promoting deep learning of content knowledge as well as developing high-end cognitive abilities, such as scientific reasoning. Emerging technology and methods should serve to facilitate (not to replace) such learning by providing more effective education settings and conveniently accessible resources. This is an important relationship that should survive many generations of technological and societal changes in the future to come. From a physicist’s point of view, a fundamental relation like this can be considered the “mechanics” of teaching and learning. Therefore, educators and researchers should hold on to these few fundamental principles without being distracted by the surfacing ripples of the world’s motion forward.

Availability of data and materials

Not applicable.

Abbreviations

American Association of Physics Teachers

Investigative Science Learning Environment

Inquiry in Scientific Thinking Analytics and Reasoning

Massive open online course

New Generation Science Standards

• Physics education research

Science Technology Engineering and Math

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The research is supported in part by NSF Awards DUE-1431908 and DUE-1712238. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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Ableism puts neurodivergent students at a disadvantage, february 29, 2024.

While undergraduate physics students that identify as neurodivergent report little outright discrimination or violence, they do say that structural ableism has negatively impacted their time as students.

Feature on: Liam G. McDermott, Nazeer A. Mosley, and Geraldine L. Cochran Phys. Rev. Phys. Educ. Res. 20 , 010111 (2024)

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Examining racial diversity and identity in physical review physics education research, july 1, 2020.

In the following special collection from Physical Review Physics Education Research , authors examine and highlight racial diversity, specifically how Black physicists and people of color navigate within the physics community at large.

Editorial: Announcing the PRPER Statistical Modeling Review Committee (SMRC)

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Lead Editor, Charles Henderson, announces PRPER’s development of the Statistical Modeling Review Committee (SMRC) to help support high-quality statistical modeling techniques.

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PRPER Lead Editor, Charles Henderson, and APS Editor in Chief, Michael Thoennessen, discuss the vital importance of offering an inclusive and welcoming environment to the physics community.

Editorial: Call for Papers Focused Collection of Physical Review Physics Education Research Instructional labs: Improving traditions and new directions

November 17, 2021.

Physics is an experimental science. Instructional laboratories where students conduct experiments, analyze data, arrive at conclusions, and communicate findings have been around for over a century. Every physics department has labs of different levels: from introductory to advanced, for majors and nonmajors, with real equipment or virtual.

Editorial: Call for Papers Focused Collection of Physical Review Physics Education Research Qualitative Methods in PER: A Critical Examination

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This Physical Review Physics Education Research (PRPER) Focused Collection was curated to bring to light curriculum design decisions and the factors that shape them. By making decisions about design explicit, we can better understand the contexts behind our research claims, hold curricula up to informed critique, and support new scholars as they undertake curriculum development.

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Physics Education Research

Physics Education Research (PER) is the study of how people learn physics and how to improve the quality of physics education. Researchers use the tools and methods of science to answer questions about physics learning that require knowledge of physics. Researchers focus on developing objective means of measuring the outcomes of educational interventions. How do we know whether our courses and interventions are successful?

One such approach is the design of diagnostic assessments and surveys. While many instructors develop questions to assess student learning, diagnostic research assessments undergo rigorous design, testing, and validation processes to facilitate objective comparisons between students and methods of instruction. These assessments are like detectors that must be carefully crafted and calibrated to ensure we understand what they are measuring.

The Cornell Physics Education Research Lab has a large focus on studying and developing learning in lab courses. Researchers are collecting data to evaluate the efficacy of lab courses in achieving various goals, from reinforcing physics concepts to fostering student attitudes and motivation to developing critical thinking and experimentation skills. They are designing innovative teaching methods to harness the affordances of lab courses, namely, working with messy data, getting hands on materials, troubleshooting equipment, and connecting physical models to the real world and data. There are many open research questions related to understanding how students learn these ideas.

This work will be facilitated by a research Active Learning Initiative grant from the Cornell University College of Arts and Sciences led by Natasha Holmes (PI). This grant will facilitate the renewal of the physics lab elements of the two calculus-based introductory physics course sequences. In addition to redesigning the instructional materials, this project will involve significant attention on understanding how instructional materials get passed down between instructors and sustained over time, how teaching assistants are trained to support the innovative designs, and many open research questions to evaluate students’ experience and learning in these courses.

The recent Cornell University Physics Initiative in Deliberate practice (CUPID) was a 5-year project to renew the introductory, calculus-based physics course sequence for Engineering and Physics majors. This project, led by Jeevak Parpia and Tomás Arias and involving more than 8 other faculty and lecturers in the department, applied results of PER to improve the teaching and learning in Cornell University courses, and to test the generalizability of results observed elsewhere. By collecting assessment, survey, and exam data across the duration of the course implementation, the group demonstrated significant improvements in student learning and attitudes. They are now in the process of monitoring how the course materials get passed on to new faculty. There are many opportunities to study differences in various forms of active learning.

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I. INTRODUCTION

Ii. cultivating teacher expertise, iii. an essental role for physicists in stem ed reform, iv. what teachers need, a role for physicists in stem education reform.

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Hestenes David; A role for physicists in STEM education reform. Am. J. Phys. 1 February 2015; 83 (2): 101–103. https://doi.org/10.1119/1.4904763

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The national crisis in K-12 STEM education is thoroughly documented, and calls are loud and clear for immediate action to maintain the status and competitiveness of the United States in the global economy. 1 Even so, education reform continues to flounder despite massive investments from the U.S. government. It is natural to wonder why.

To serve as an authoritative guide for deep and coherent STEM education reform, the National Research Council (NRC) has recently published A Framework for K-12 Science Education . 2 The “Framework” was developed by a distinguished committee of the NRC led by physicist Helen Quinn. They did a masterful job of updating previous recommendations and broadening them to include engineering and technology, with a balanced emphasis on scientific inquiry and engineering design.

As a guide for implementing the Framework, the Next Generation Science Standards (NGSS) 3 was developed collaboratively with states and other stakeholders in science education and industry in a process led by a nonprofit organization called Achieve. 4 Progress has stalled since then. Many states have been slow to adopt the NGSS, while others created committees to pick it apart and, in the name of accountability, establish a hodgepodge of hurdles and hoops for schools to prove that they are teaching the students something. The details are left to the schools and school districts.

The outcome can be predicted with certainty. For we have already seen how the last generation of National Science Education Standards 5 was diluted and butchered almost beyond recognition at state and local levels. Typically, the schools solve the problem of STEM education by ordering textbooks from publishers who carefully note how each particular standard is covered in their books. Too often, teaching is reduced to passing out textbooks and assignments, followed by drill and practice with hope for a good return on mandated exams. Too seldom, the principal encourages and supports independent/innovative initiatives by the teacher.

How could it be otherwise? Schools and school districts are ill-equipped to implement reforms and conduct the necessary professional development on their own, because they lack the necessary expertise in science and technology as well as the resources to keep up-to-date with advances in science curriculum materials and pedagogy. Nor can we expect reform from schools of education. The education establishment is too slow and ponderous to cope with the rapid evolution of the STEM disciplines, and responsibility is too broadly distributed for decisive action.

Fortunately, there is a better way. A new approach to STEM education reform that has emerged from the physics community and will depend on its support for continued success. My aim here is to argue that certain elements of this approach are essential for ultimate reform. This is not to deny the value of other approaches; indeed, diversity is welcome, and continued progress will require contributions from across the scientific community. But I am offering a perspective drawn from a program that I have been intimately involved with for more than two decades. Only a brief description of the origin, state, and potential of this new approach to broad STEM education reform is possible in this editorial. More details are given elsewhere. 6,7

The serious problem of pre-service preparation for physics teaching has been thoroughly addressed in a recent editorial in this journal, 8 and I fully concur with their assessment. But pre-service preparation is only one side of the coin; the other side is in-service professional development. I am not the only one who thinks that such continuing professional development is important. As the prestigious Glenn Commission 9 concluded:

“We are of one mind in our belief that the way to interest children in mathematics and science is through teachers who are not only enthusiastic about their subjects, but who are also steeped in their disciplines and who have the professional training—as teachers—to teach those subjects well. Nor is this teacher training simply a matter of preparation; it depends just as much—or even more—on sustained, high-quality professional development.”

This conclusion is abundantly supported by the remarkable success of the Modeling Instruction Program over the last 24 years. Since I have been involved from the beginning, I can give you an insiders view of its evolution. The program is grounded in a decade of physics education research that led to the development of a scientific pedagogy called Modeling Instruction , so named because it takes the creation and use of conceptual models as the core of scientific practice. Modeling Instruction was then embedded in a series of summer Modeling Workshops on teaching introductory physics, which began in 1990. Over the next fifteen years, with support from the National Science Foundation (NSF), Modeling Instruction evolved into a national program for physics teaching reform. To date, more than 3000 physics teachers—10% of physics teachers in the U.S.—have taken a Modeling Workshop.

Many participants in the physics Modeling Workshops were “crossover teachers” from chemistry, biology, and mathematics. They were so impressed with the pedagogy that they demanded similar workshops in their fields. Thus, the teachers themselves took the lead in developing Modeling Workshops in chemistry, physical science, and biology, and they are well on their way to creating a fully integrated set of STEM curriculum materials in full accord with the NRC Framework for science education. To my knowledge, no other program is half so far along in curricular development—and this by teachers themselves without significant external funding.

In 2005, when NSF funding for Modeling Instruction ceased, the teachers created their own organization, the American Modeling Teachers Association (AMTA), 10 to keep the program going. To date, the AMTA has more than 2000 members, and coordinates 80 Modeling Workshops for science teachers each summer. Approximately 7500 teachers have attended one or more Modeling workshops, and of these teachers, almost 6000 of them are still subscribed to one or more of seven Modeling listservs.

I used to worry about the fact that some two thirds of high school physics teachers do not have even an undergraduate minor in physics, and the pipeline for new teachers can hardly keep up with the retirement rate. 8 But, in my years directing the Modeling Program, I have seen that many crossover teachers do a much better job teaching physics than most physics majors. I attribute this partly to their embrace of Modeling pedagogy and partly to the intellectual curiosity and self-confidence of those who venture to crossover teaching. Outcomes have convinced me that training in summer Modeling Workshops is sufficient to develop any capable individual with some background in science or engineering into a competent STEM teacher—especially with the continued support and fellowship provided by the AMTA.

To my utter surprise and delight, the Modeling Program has morphed into a large, cohesive community of practice with a common vision of effective STEM teaching. Though I helped get it started, the teachers did the heavy lifting, and the AMTA is prepared to marshal the teachers in a collaborative effort to reform STEM education across the board. It is time now for the community of scientists, especially physicists, to step up in support.

Physicists know that physics must play a leading role in STEM education. This has been expressed explicitly in the American Association of Physics Teacher (AAPT) advocacy of Physics First , 11 a reorganization of high school science that puts physics in 9th grade, before chemistry and biology. Opponents claim that physics is too difficult for most students and point to failures in implementing Physics First. On the contrary, the AMTA has decisive evidence 12–14 of successful implementation with Modeling pedagogy and clear benefits for STEM courses that follow.

Another successful approach called Physics Union Mathematics (PUM) 15 expressly addresses the important problem of integrating physics with mathematics at the 9th grade level. It draws from the Investigative Science Learning Environment 16 for introductory physics developed by Eugenia Etkina and Alan Van Heuvelen, which is fully compatible with the Modeling approach. A common ingredient of these successful courses is grounding in Physics Education Research. Besides serving as the keystone course for the ideal of physics for all citizens, 9th grade physics is the lynchpin for unifying the entire STEM curriculum.

There are also deep psychological reasons why physics should play a central role in the STEM curriculum. It is no accident that physics and astronomy were the first sciences to develop historically. Physics is the science most closely related to our basic perceptions of matter, motion, and light. The science of force and motion should be taught first, because it relates directly to the students sensory experience. Furthermore, physics provides the foundation for quantitative methods in the rest of science, and it stands as the first exemplar of scientific method. Quantitative reasoning with number and unit goes hand-in-hand with modeling and measurement, which couples mathematics to science.

For nationwide implementation of STEM education reform, the AMTA needs strong support from the physics community. Reform can move rapidly forward without waiting for new funding because the AMTA already has a stable of accomplished teachers ready to move, and only physicists have the resources at hand to support them. In particular, PhysTEC, 17 a partnership between the American Physical Society (APS) and the AAPT to help a coalition of more than 300 universities upgrade their physics teacher education programs.

PhysTEC has concentrated on pre-service teacher education and recently engaged the AMTA to give an online course on modeling pedagogy for coalition members. 18 As a natural next step, PhysTEC should be encouraged to hold sessions on building in-service programs at its conferences. For any university committed to serving the educational needs of its local community or region, the AMTA is prepared to guide development of a Local Teacher Alliance (LTA) of STEM teachers and link it to the national AMTA network for teacher enhancement and STEM education reform. All that is needed to start is sponsorship from the physics department. There are too many details about establishing a thriving LTA to discuss here; 6 suffice it to say that the AMTA is already linked to high-functioning LTAs scattered across the nation. The good news is that these LTAs have been created and run by the teachers alone. The bad news is that in most cases, local universities have not learned of the great advantages in linking up with them. For starters, the LTA can provide a direct pipeline of students from high school to the STEM disciplines in college, and such links are prerequisites for successful STEM education reform.

The proposal that physicists must take the lead in organizing scientists and engineers to support STEM education reform may seem gratuitous, but the fact is that other disciplines are not nearly so well prepared to do it. Consider chemistry, for example. Though the AAPT has been supporting physics teachers for the better part of a century, the American Chemical Society (ACS) created the comparable American Association of Chemistry Teachers (AACT) only recently. 19 Making up for lost ground, the ACS is also partnering with APS to create a ChemTEC to link up with PhysTEC, and the AACT is discussing collaboration with the AMTA. Already, some 400 chemistry teachers are taking Chemistry Modeling Workshops each year. Likewise, links of the physics department to other STEM disciplines provide a natural pathway to involve them in supporting LTAs for STEM teachers.

The expertise of a master teacher and the effort needed to acquire it is vastly underestimated by nearly everyone. Development of expertise in any domain requires approximately ten years (or 10,000 h) of “deliberate practice” specifically aimed at improving performance. 20,21 This applies as much to physics teachers as to physicists. But look at the disparity in their opportunities. Teachers are thrown into the classroom immediately after college graduation, whereas physicists typically have another five to seven years of graduate and postdoctoral work before they become independent scientists. To make up the difference, teachers need access to lifelong professional development like that provided by the AMTA in Modeling Workshops. University science and engineering departments must be involved in helping to supply it. 7  

Empowering teachers is the key to STEM education reform, and STEM teachers need resources and support more than accountability. More than anything, teachers need to be integrated into the community of scientists as respected colleagues in ways that strengthen their expertise, their credibility, and their impact in the schools.

Teachers themselves should be the local experts on the STEM curriculum and how to teach it. They should be advising their principals and school districts on what needs to be done, rather than the other way around. To command that authority, teachers need the direct support from the scientific community that connection to a university and a Local Teacher Alliance can provide. Only then can they freely implement up-to-date STEM education in their schools. To be sure, many districts and principals will not readily yield their prerogatives even to teachers and curricula that have endorsements from the broader scientific community. But I know of many who are eager for it, and their successful engagement will stimulate others to follow. Reform is for the long haul.

Physicists must take the lead in bringing all this to pass, because they have unique access to resources for building the necessary infrastructure. As I have discussed, a crucial first step for a physics department is to establish an LTA linked to the AMTA. Then colleagues from other disciplines can be invited to join in expanding the LTA from physics to a larger STEM LTA or a coalition of LTAs for different disciplines.

With their direct connections to STEM teachers and students, LTAs are ideal vehicles for university Outreach Programs as well as sponsoring student symposia, summits, and retreats. In short, LTAs can be powerful mechanisms for bonding universities to their communities.

[Hestenes is cofounder of Modeling Instruction. For this work he received the 2002 Oersted Medal from the American Association of Physics Teachers, the 2003 Education Research Award from the Council of Scientific Society Presidents, and shared the 2014 APS Excellence in Physics Education Award.]

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• Published: 01 February 2023

How to restore trust in science through education

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The authority of science within society is contested by antiscientific movements. To restore trust, science education should involve students in the social processes of knowledge production.

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Formal and informal learning in science: the role of the interactive science centres

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The Roles of Physics in Our Modern Society

Difference Between Metaphysics & Quantum Physics

Physics touches every aspect of our lives. It involves the study of matter, energy and their interactions. As such, it is one area of science that cuts across all other subjects. Other sciences are reliant on the concepts and techniques developed through physics. Other disciplines — such as chemistry, agriculture, environmental and biological sciences — use the laws of physics to better understand the nature of their own studies. Physics focuses on the general nature of the natural world, generally through a mathematical analysis.

Public Interest in Physics

Physics is one of the most difficult subjects taught in schools. A number of students are even more daunted with its use of mathematics. In a study done in UK from 1985 to 2006, it was found that there was 41 percent decrease in the number of entries to A-level examinations in sciences. This decreasing trend is similar in other countries. Despite this trend, physics remains an integral part of the educational system. It is through physics that new methodologies were developed that helped improve the quality of life, including things such as automobiles and modern construction.

Importance of Physics in the Current Society

Society’s reliance on technology represents the importance of physics in daily life. Many aspects of modern society would not have been possible without the important scientific discoveries made in the past. These discoveries became the foundation on which current technologies were developed. Discoveries such as magnetism, electricity, conductors and others made modern conveniences, such as television, computers, phones and other business and home technologies possible. Modern means of transportation, such as aircraft and telecommunications, have drawn people across the world closer together — all relying on concepts in physics.

Importance of Physics in Meeting Future Energy Requirements

In 1999 during the World Conference on Science (WCS), the UNESCO-Physics Action Council considered physics an important factor in developing solutions to both energy and environmental problems. Physics seeks to find alternative solutions to the energy crisis experienced by both first world and developing nations. As physics help the fields of engineering, bio-chemistry and computer science, professionals and scientists develop new ways of harnessing preexisting energy sources and utilizing new ones.

Importance of Physics in Economic Development

In the United Nations Millennium Summit held in 2000, it was recognized that physics and the sciences will play a crucial role in attaining sustainable development. Physics helps in maintaining and developing stable economic growth since it offers new technological advances in the fields of engineering, computer science and even biomedical studies. These fields play a crucial role on the economic aspect of countries and finding new and better ways to produce and develop products in these fields can help boost a country’s economy. Similarly, the International Union of Pure and Applied Physics (IUPAP) asserted that physics will generate the necessary knowledge that will lead in the development of engines to drive the world’s economies.

In Rwanda, the education ministry was mandated to develop the country’s scientific and technical know-how. Medical physics and information technology benefited the country by developing a national nutrition program and an epidemic surveillance system. Physics and engineering helped rural areas gain safe drinking water through gravimetric techniques, irrigation techniques and rainwater harvesting.

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How do airplanes fly? – Benson, age 10, Rockford, Michigan

Airplane flight is one of the most significant technological achievements of the 20th century. The invention of the airplane allows people to travel from one side of the planet to the other in less than a day, compared with weeks of travel by boat and train.

Understanding precisely why airplanes fly is an ongoing challenge for aerospace engineers, like me , who study and design airplanes, rockets, satellites, helicopters and space capsules.

Our job is to make sure that flying through the air or in space is safe and reliable, by using tools and ideas from science and mathematics, like computer simulations and experiments.

Because of that work, flying in an airplane is the safest way to travel – safer than cars, buses, trains or boats. But although aerospace engineers design aircraft that are stunningly sophisticated, you might be surprised to learn there are still some details about the physics of flight that we don’t fully understand.

May the force(s) be with you

There are four forces that aerospace engineers consider when designing an airplane: weight, thrust, drag and lift. Engineers use these forces to help design the shape of the airplane, the size of the wings, and figure out how many passengers the airplane can carry.

For example, when an airplane takes off, the thrust must be greater than the drag, and the lift must be greater than the weight. If you watch an airplane take off, you’ll see the wings change shape using flaps from the back of the wings. The flaps help make more lift, but they also make more drag, so a powerful engine is necessary to create more thrust.

When the airplane is high enough and is cruising to your destination, lift needs to balance the weight, and the thrust needs to balance the drag. So the pilot pulls the flaps in and can set the engine to produce less power.

That said, let’s define what force means. According to Newton’s Second Law , a force is a mass multiplied by an acceleration, or F = ma.

A force that everyone encounters every day is the force of gravity , which keeps us on the ground. When you get weighed at the doctor’s office, they’re actually measuring the amount of force that your body applies to the scale. When your weight is given in pounds, that is a measure of force.

While an airplane is flying, gravity is pulling the airplane down. That force is the weight of the airplane.

But its engines push the airplane forward because they create a force called thrust . The engines pull in air, which has mass, and quickly push that air out of the back of the engine – so there’s a mass multiplied by an acceleration.

According to Newton’s Third Law , for every action there’s an equal and opposite reaction. When the air rushes out the back of the engines, there is a reaction force that pushes the airplane forward – that’s called thrust.

As the airplane flies through the air, the shape of the airplane pushes air out of the way. Again, by Newton’s Third Law, this air pushes back, which leads to drag .

You can experience something similar to drag when swimming. Paddle through a pool, and your arms and feet provide thrust. Stop paddling, and you will keep moving forward because you have mass, but you will slow down. The reason that you slow down is that the water is pushing back on you – that’s drag.

Understanding lift

Lift is more complicated than the other forces of weight, thrust and drag. It’s created by the wings of an airplane, and the shape of the wing is critical; that shape is known as an airfoil . Basically it means the top and bottom of the wing are curved, although the shapes of the curves can be different from each other.

As air flows around the airfoil, it creates pressure – a force spread out over a large area. Lower pressure is created on the top of the airfoil compared to the pressure on the bottom. Or to look at it another way, air travels faster over the top of the airfoil than beneath.

Understanding why the pressure and speeds are different on the top and the bottom is critical to understand lift . By improving our understanding of lift, engineers can design more fuel-efficient airplanes and give passengers more comfortable flights.

The conundrum

The reason why air moves at different speeds around an airfoil remains mysterious, and scientists are still investigating this question.

Aerospace engineers have measured these pressures on a wing in both wind tunnel experiments and during flight. We can create models of different wings to predict if they will fly well. We can also change lift by changing a wing’s shape to create airplanes that fly for long distances or fly very fast.

Even though we still don’t fully know why lift happens, aerospace engineers work with mathematical equations that recreate the different speeds on the top and bottom of the airfoil. Those equations describe a process known as circulation .

Circulation provides aerospace engineers with a way to model what happens around a wing even if we do not completely understand why it happens. In other words, through the use of math and science, we are able to build airplanes that are safe and efficient, even if we don’t completely understand the process behind why it works.

Ultimately, if aerospace engineers can figure out why the air flows at different speeds depending on which side of the wing it’s on, we can design airplanes that use less fuel and pollute less.

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Modelling Roles of Mathematics in Physics

Perspectives for Physics Education

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• Published: 07 November 2022
• Volume 33 , pages 365–382, ( 2024 )

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Modelling roles of mathematics in physics has proved to be a difficult task, with previous models of the interplay between the two disciplines mainly focusing on mathematical modelling and problem solving. However, to convey a realistic view of physics as a field of science to our students, we need to do more than train them to become fluent in modelling and problem solving. In this article, we present a new characterisation of the roles mathematics plays in physics and physics education, taking as a premise that mathematics serves as a constitutive structure in physics analogous to language. In doing so, we aim to highlight how mathematics affects the way we conceptualise physical phenomena. To contextualise our characterisation, we examine some of the existing models and discuss aspects of the interplay between physics and mathematics that are missing in them. We then show how these aspects are incorporated in our characterisation in which mathematics serves as a foundation upon which physical theories are built, and on which we may build mathematical representations of physical information that in turn serve as a basis for further reasoning and modifications. Through reasoning processes mathematics also aids in generating new information and explanations. We have elucidated each of these roles with an example from the historical development of quantum physics. To conclude, we discuss how our new characterisation may aid the development of physics education and physics education research.

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An Educational Perspective on the Connections Between Physics and Mathematics

Physics teaching: mathematics as an epistemological tool.

Fabiana B. Kneubil & Manoel R. Robilotta

Physical–Mathematical Modelling and Its Role in Learning Physics

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1 Introduction

The relationship between physics and mathematics has been widely discussed in the field of physics education research. It is undeniable that mathematics has an important, even inseparable, role in physics, and thus also in physics education. However, modelling the role of mathematics in physics has turned out difficult, and the focus has most often been quite narrow. The existing literature has largely concentrated on problem solving through mathematical manipulations or on mathematical modelling. In physics education research, student reasoning has often been studied using thinking-out-loud problem-solving interviews, e.g. in Walsh et al. ( 2007 ), Bing and Redish ( 2009 ), and Kuo et al. ( 2013 ), just to mention some. Also models and their significance have been widely discussed in many areas. For example Greca and Moreira ( 2001 ) and Kanderakis ( 2016 ) provide some notable discussions on relationships among physical and mathematical models and how they consequently affect scientific understanding.

Arguably all these contributions build on the general issue of the role of mathematics in physics. Recently, Pospiech ( 2019 ) has provided an extensive overview of previous research on the topic, and they present multiple models that intend to capture the interplay between physics and mathematics. These models describe how the roles of mathematics manifest in physics, again emphasising the aforementioned perspectives of mathematical modelling and problem solving. Typically, their starting point is a qualitative description of a system or phenomenon (e.g. a problem assignment) that is then translated into a mathematical form using available mathematical tools. As the existing models highlight, this process generally requires physical understanding in addition to proficiency with mathematical methods. The mathematised description of the system or phenomenon can subsequently be manipulated mathematically and finally reinterpreted in physical terms. Our goal in this paper is to examine and build on this perspective by suggesting a novel characterisation of the roles of mathematics in physics, and by illustrating our characterisation with examples from the history of quantum physics. In addition, we discuss the implications of our characterisation to physics education, especially at the university level.

In this article, we have chosen for discussion two of the most prominent theoretical models of the interplay presented by Pospiech ( 2019 ): the modelling cycle adapted from mathematics education research by Redish and Kuo ( 2015 ) (see also, Redish, 2005 ; Redish & Smith, 2008 ) and the revised, more nuanced version of this model by Uhden et al. ( 2012 ). These models provide for our discussion a sufficient background into how the use of mathematics in physics is conceptualised in many of the models presented in physics education research. However, while the models by Redish and Kuo ( 2015 ) and Uhden et al. ( 2012 ) mostly focus on physics in high school or early university levels, our characterisation intends to describe the roles of mathematics in general. Thus, its implications for physics instruction can, at least in principle, be extended to all levels of education.

Our aim is to discuss what aspects of the interplay between physics and mathematics are missing in the previous models and how to incorporate them into a new kind of characterisation of the roles mathematics plays in physics. To achieve this aim, we have chosen to distance ourselves from the modelling and problem-solving perspective, and instead emphasise how mathematics affects the way we conceptualise physical phenomena. We do not intend to discuss only how mathematics is utilised in classrooms, but also probe the epistemological foundations of physics knowledge: how physics knowledge is formed and what is the role of mathematics in this process. Rooting our discussion in existing literature, we discuss the relationship between physics and mathematics from the point of view previously presented by, e.g. Pietrocola ( 2002 ) and Kneubil and Robilotta ( 2015 ), namely that contemporary physics has largely taken form within an interrelationship with mathematics. In a sense, mathematics has given physicists an ability to think about the material world (Pietrocola, 2002 ). Because of this the way, we conceptualise physical phenomena is often characteristically mathematical. Of course, this issue touches on some fundamental questions on the epistemology of mathematics (see, e.g. Quale, 2011a , b ), which in turn may stem from unresolved philosophical debates on the nature of knowledge and “realness”. However, our view is that the interplay of physics and mathematics can be characterised meaningfully without an explicit commitment to any certain ontology of mathematics, and hope our article also serves to demonstrates this approach.

Mathematics accounts for much of the form of physical knowledge. This is to say that the mathematical constructs available at the time of formation of a physical theory contribute to shaping the theory. On the other hand, the mathematical constructs used are also selected and modified for the use of a particular theory and its physical context. This idea of a two-way dynamic between mathematical and physical constructs and knowledge is echoed by the evolutionary perspective of human knowledge presented by Galili ( 2018 ). In essence, this view posits that the effectiveness of mathematics in accounting for reality is a result of a long process in which mathematical constructs have been tested in (or conceived for) the construction of physics knowledge. The failed attempts have been discarded while successful constructs have found their place in established structures.

As was earlier asserted, following the argument by Kneubil and Robilotta ( 2015 ), physics has been formed in close connection with mathematics. Naturally this entails that the relationship between the two disciplines has not been one-sided, mathematics only feeding information to physics. On the contrary, physics has played a significant role in the advancement of mathematics. For example Kjeldsen and Lützen ( 2015 ) have discussed how physics contributed to the formation of the concept of mathematical function, just to mention one example. This complex interaction between the disciplines has been extensively discussed in Galili ( 2018 ). However, in this article we have limited our perspective to the role mathematics plays in physics, and we merely touch on the issue of their other interactions.

Of course, recognition of these issues is not new to science education. For example in the Science for All Americans framework (AAAS, 1989 , p. 17), mathematics “provides science with powerful tools to use in analysing data” and “is the chief language of science” that provides precision and a “grammar” for analysis. Accordingly, the mathematical underpinnings of scientific activity and knowledge has seen some discussion in nature of science (NOS) literature (see e.g. Erduran et al., 2019 ; Galili, 2019 ), even if it arguably has rarely been at the forefront. However, here we have focused on a historical-philosophical perspective, similarly to, e.g. Uhden et al. ( 2012 ). By stepping back from a problem-solving centred perspective, we also direct our discussion primarily to higher education, but will suggest some educational implications relevant for lower educational stages as well.

Indeed, from all these perspectives, the role of mathematics in physics is much more than serving as a tool for modelling and manipulations, and our main goal in this article is to illustrate this further. First and foremost, in our new characterisation we argue that mathematics can be regarded as providing physical theories with a skeletal structure. This is the foundation upon which we may build various representations (models) following the physical–mathematical rules stemming from the underlying structure. These representations in turn can be shared (communicated) and examined further. Further reasoning based on a representation may give rise to new insights that would not have been possible were the situation not presented using mathematics. These processes often get trivialised if we focus only on mathematisation and manipulations.

In the following, we have elucidated each presented role with an example from the historical development of quantum physics. We have selected quantum physics as our reference theory because it is relatively easy to find suitable examples from it. This is likely because quantum physics is a highly math-intensive area of physics. However, we could have chosen the examples from, e.g. electro-magnetism that has previously been the subject of extensive discussion in similar contexts (cf. Silva, 2007 ; Tweney, 2009 , 2011 ).

While the roles explicated in our new characterisation may be new to previous models that have aimed to depict the interplay between physics and mathematics, they have been discussed in other contexts. For example Gire and Price ( 2015 ) have examined the significance of notational systems on problem solving, while Tweney ( 2009 , 2011 ) has discussed the contribution of different representations to the subsequent reasoning processes. Various ways of mathematical reasoning have been discussed, e.g. by Hull et al. ( 2013 ) who aim to shift the focus to more blended reasoning in problem solving, and Tzanakis and Thomaidis ( 2000 ), Pask ( 2003 ), Silva ( 2007 ), and Gingras ( 2015 ) who have examined the use of analogies and deductive as well as inductive reasoning. The way mathematics gives rise to new knowledge has also been discussed elsewhere: for example Quale ( 2011a , b ) has examined the appearance of unexpected entities and solutions, and Vemulapalli and Byerly ( 2004 ) have focused on quantitative variables in physical theories. The purpose of our new characterisation is to collect and organise these existing but somewhat scattered pieces of the puzzle, and in doing so suggest a background for the discussion on the roles mathematics plays in physics learning and teaching.

In the next sections, we first present a brief overview of the existing models, discuss the elements missing in these models and then present a new characterisation looking at the theme from a different angle. We begin by presenting two previous depictions of how mathematics is used in physics. After this, we present our characterisation of the roles mathematics takes in different situations in the context of physics. Each role is demonstrated with an example from the history of quantum physics. The existing models and the proposed characterisation are then contrasted with each other and with the previous research literature.

2 Models of the Interplay

Pospiech ( 2019 ) has reviewed the topic of mathematics in physics education from many perspectives. They discuss, for example the historical-philosophical perspective, conceptual understanding, and external representations. Most notably for the present article, they review multiple models of the interplay between mathematics and physics from the physics education point of view.

In the next sections, we follow Pospiech’s ( 2019 ) discussion of theoretical models of mathematisation in physics. Key to these models is to focus on translating physical elements into mathematics and vice versa. We note, however, that we will not discuss models of the interplay not essential for our focus.

2.1 Modelling Cycle

Redish and Kuo ( 2015 ) consider “the language of mathematics in physics” from the perspective of cognitive linguistics: they argue that “in science, we don’t just use math, we make meaning with it in a different way than mathematicians do” (Redish & Kuo, 2015 , p. 561). They base this discussion on the notion that the use of mathematics in physics is vastly different from how mathematicians use mathematics. According to them, this is because physicists use mathematics to describe, learn about, and understand physical systems, while for mathematicians, mathematical expressions are free from the ancillary (and often implicit, tacit, or unstated) physical meaning of symbols. To explicate their view of how mathematics is used in modelling and problem solving in physics classes, they present the diagram in Fig.  1 , called a model of mathematical modelling or a modelling cycle (see also, Redish, 2005 ; Redish & Smith, 2008 ), and describe the modelling process having the following steps:

The modelling process begins by identifying the physical system to be modelled (the lower left corner of the diagram). The system is defined and isolated from its environment, and the quantifiable variables and parameters of the system are identified.

By choosing the mathematical structures appropriate for describing the features of the system, it becomes possible to map the measures of the system onto mathematical symbols. This step is called modelling. The resulting mathematical representation of the physical system is thus composed of the measures of the physical system and the relations between them.

As the mathematical representation of the physical system inherits the features and rules confining the chosen mathematical structures, it now becomes possible to process the modelled system mathematically. By mathematical manipulations, mathematical results are achieved.

The mathematical results must be interpreted back into physics in order to make them understandable in the context of physics. The interpreted results are finally compared with the original physical system. This step is called evaluation.

Modelling cycle by Redish and Kuo ( 2015 )

Redish and Kuo ( 2015 ) emphasise that in reality the use of mathematics in physics is not as simple or sequential as their diagram indicates. They note that the diagram does not intend to capture this entanglement of disciplines but “to emphasise that our traditional way of thinking about using math in physics classes may not give enough emphasis to the critical elements of modelling, interpreting, and evaluating” (p. 568). As such, the modelling cycle parallels models previously presented in mathematics education research, depicting the translational processes between the real world and mathematics (cf. Blum & Leiß, 2005 ). In addition, its clarity and simplicity make it a useful step in introducing more complex models.

Although the educational level on which the model by Redish and Kuo ( 2015 ) is intended to be utilised is not specified, the examples presented in their article seem to be of early university level. The model is indeed very well suited for solving traditional introductory and upper division level exercise problems.

2.2 Revised Modelling Cycle

Uhden et al. ( 2012 ) have noted some shortcomings in the modelling cycle and other previous models. According to their criticism, the modelling cycle conceptualisation does not address different levels or “degrees” of mathematisation. Furthermore, and crucially for education, it does not clearly distinguish between the technical and structural role of mathematics that they define after Pietrocola ( 2008 ): the technical role refers to the algorithmic use of mathematics, e.g. rote manipulations, while the structural role is related to the more conceptual use which is more “entangled” with physics. In education, these two roles map onto definitions of technical and structural skills, where “technical skills are associated with pure mathematical manipulations whereas the structural skills are related to the capacity of employing mathematical knowledge for structuring physical situations” (Uhden et al., 2012 , p. 493).

To address these issues, Uhden et al. ( 2012 ) propose an alternative model, a revised modelling cycle, shown in Fig.  2 . In this model, the purely qualitative physics is accounted for by the lowest level of the physical–mathematical model, with each of the higher levels representing the model at a new degree of mathematisation. Pure mathematics is detached from the physical–mathematical model. They describe the steps of mathematical reasoning as follows:

Arrow (1) stands for the translation from the world to the physical–mathematical model, i.e. the idealisation process. Just as in the modelling cycle by Redish and Kuo ( 2015 ), an idealised model of a physical system is constructed.

In the modelling or problem-solving process, we move up and down in the diagram as we carry out mathematisation and interpretation phases. In the diagram, this is denoted by arrows (2) and (3), respectively. In practice, this may mean refining our mathematical model (moving towards a more mathematised representation) and interpreting the physical meaning of mathematical expressions (moving downwards on the latter of mathematisation). The mathematisation and interpretation steps (arrows (2) and (3)) are identified as application of structural skills whereas arrow (4), making a round to the pure mathematics, represents technical skills, e.g. doing calculations. Uhden et al., ( 2012 , p. 498) describe this step as “doing just mathematics”.

Arrow (5), standing for the validation process in which results are interpreted back into the world, closes the cycle.

Revised modelling cycle by Uhden et al. ( 2012 )

Uhden et al. ( 2012 ) highlight that their model is intended to be used as a diagnostic tool for mathematical reasoning in physics: for example the model may help in distinguishing between the structural and technical skills required in problem-solving contexts. Ultimately, they envisage the model being used as an instrument to support teaching strategies which focus on the structural role of mathematics.

Uhden et al. ( 2012 ) mention that their discussion moves mainly on high school and university levels. Also their chosen example for the application of the model, the classical physics problem of free fall, may be treated both in high school and in an introductory university course on different levels of abstraction. However, they emphasise, their discussion is applicable to physics education in general.

Even though the revised modelling cycle is indeed able to capture a more detailed depiction of the use of mathematics in physics, it still overlooks some subtleties that arise from the entanglement of physics and mathematics. To highlight these aspects, we have constructed another characterisation of these roles in physics and physics education that we will present in the next section.

3 New Characterisation of the Interplay

In this section, we present a new characterisation of how the roles of mathematics manifest in physics and physics education. The characterisation does not intend to provide yet another visual, cyclic depiction of the actions (and skills) that involve physical–mathematical reasoning. Instead, we aim to dissect and analyse the roles more deeply and to describe, through examples from the history of quantum physics, how they manifest. In this sense, our characterisation is meant to answer somewhat different questions than the previously presented models: the modelling cycles focus on modelling and problem-solving actions and skills, whereas our characterisation aims to provide a more general description of the roles mathematics takes in physics (hence our use of the word “characterisation”).

Moreover, our characterisation aims to describe the roles of mathematics outside some specific type of activity on some stage of physics education, thus being relevant all the way up to the graduate level of universities. As the scope of the characterisation is wider, its contents are also more general and abstract than those of the modelling cycles. However, we argue, it can still be applied on lower educational levels in a simplified form. For example ideas of utilising underlying rules in forming representations or carrying out calculations resemble closely the actions presented in the modelling cycles. Furthermore, the characterisation provides vocabulary for discussing, even somewhat practically, the role of mathematics in (the nature of) science; for example the case of the positron (see Section  3.4.1  below) should spark valuable classroom dialogue even with younger students.

We begin presenting the new characterisation by discussing mathematics as a foundation upon which physical theories are built. Then we proceed to examine how mathematics contributes to representations of physical information that in turn serve as a basis for further reasoning and modifications. Lastly, we review the ways in which mathematics aids in generating new information and explanations. Each of these roles is presented in their own section followed by an illustrating example from the history of quantum physics. Note that the selected examples do not cover all the facets of the roles presented in the characterisation. We have selected them because they exemplify some central features of the characterisation.

A summary of the roles in our characterisation is presented in Table 1 . This table is intended to support understanding the characterisation as a whole. In the following text, we have used numbering to identify the subroles presented in the table when they appear for the first time. However, it should be noted that both in the table and in the following descriptions, the limits of different roles of the characterisation are not clear-cut. The roles are mutually dependent (for example syntactic structure affects representations) and thus there naturally is some overlap.

3.1 Syntactic Structure

Starting with the fundamentals, mathematics has a role in forming the basic structure that fixes the rules and limitations for physical theories. More precisely, mathematics aids in forming a syntactic structure for physics: it acts as a framework in which physical knowledge is organised (Table 1 , role 1a). We call this structure syntactic as it parallels the idea of linguistic syntax in the sense that it provides the form of physical theories (1b). For a natural language, the rules and limitations stem from, e.g. grammar and vocabulary that define how the language can be used and what can be expressed using it. Syntactic structure also limits the scope of things that can be perceived in terms of the language, i.e. what kind of entities and structures can be made understandable using it.

There exists an organised collection of mathematically expressed information that is thought to be valid in physics at a given moment. The syntactic structure organising this information is formed by the core of mathematical axiomatic principles, on top of which physical–mathematical theories are built. Here we mostly discuss the mathematical nature of this structure, but it is worth noting that we do not intend to say that it is thoroughly mathematical. On the contrary, a physical theory is a collection of physical information that is organised and expressed mathematically. On the same note, purely syntactic bits of the structure form its hard nucleus, while the higher layers bring also semantic aspects into it. The semantics is further discussed in the next section.

The established theory structure provides rules and limits for actions in physics. These rules affect many things from planning experiments, representing physical systems, mathematical manipulations and logical reasoning to interpretation of results (1d–1f). The rules stem from both physical and mathematical aspects of the syntactic structure, and they can be either physical or mathematical in nature. For example following mathematical rules guarantees that the end result is logically true. Logic does not, however, ensure that the result is physically possible because logically consistent results can be ruled out by the selected physical background theory or the context. After all, while, e.g. solving a physical problem, we may end up with multiple mathematical results, from which we have to choose the plausible one(s) in the given context. These cases bring our attention to the interface of mathematical and physical–mathematical structure: physical solutions are ones that do not violate theoretical assumptions by, e.g. exceeding the speed of light (1 g).

Even though the theory structure is established, it can still be modified. For example the advancement of scientific knowledge sometimes requires incorporating previously unapplied mathematics into the theory structure, while some new observations may even shake the foundations of the existing structures. Indeed, one purpose of scientific research is to study the soundness of existing theories, to test the limits of theories and expand their reach to new domains. New results, if they are found to be reliable enough, are attached as a part of the theory structure (1c). This image of syntactic theory structures closely resembles Kuhn’s ( 1962 ) paradigms of normal science.

3.1.1 Example: the Bohr Atomic Model

The history of quantum physics is a story about realisations of the shortcomings of classical physics and the inception of new quantum–mechanical theories that required physicists to adopt new mathematical theories into their repertoire. As an example of this development, let us consider the conception of the Bohr atomic model.

At the turn of the nineteenth and twentieth centuries, it was realised that atoms are not the smallest possible constituents of matter. In the subsequent years, multiple models describing the subatomic structures were developed, most famously the models by J. J. Thompson and Rutherford. Bohr was very familiar with the development of such atomic models, as around the time he first worked with Thompson in Cambridge and then moved to work with Rutherford in Manchester (Jammer, 1989 ). According to Jammer ( 1989 ), as the first step of the work that finally led to the new atomic model, Bohr investigated to what extent classical mechanics and electrodynamics (the main body of the theory structure of the time) could account for Rutherford’s model that assumes electrons revolving around a massive, positively charged nucleus. Based on these examinations, he soon understood that the stability of the Rutherford atom cannot possibly be explained in classical terms. Thus, he made the assumption that only certain discrete electron orbits are allowed, and by incorporating Planck’s quanta into this model, he was able to find the right numeric solution for the hydrogen energy spectrum.

Building on nineteenth century physics and the theory structure of that time, Bohr’s work extended the reach of the existing syntactic theory structure. He utilised his knowledge of the classical theories of mechanics and electrodynamics as well as the theoretical ideas and observational results about the atomic structure presented by others before him. Combining these pieces of the puzzle with a few bold assumptions, Bohr was able to build a new atomic model that in turn became a milestone on the way towards the full-fledged quantum theory.

3.2 Representations and Semantics

While the syntactic structure provides a framework in which physical knowledge is organised, there is another role mathematics has in expressing this knowledge: in addition to the syntactic mathematical structure giving the rules and limits for representations, mathematics also provides their shape (Table 1 , role 2a). We call this level of representations semantic because of its resemblance to linguistic semantics: constructing a representation loads the used syntactic structures with meaning. The objects we choose to use in our representation are given meaning in relation to each other, and to their physical background theory and context. This is quite analogous to the way we make meaning in natural languages. For example the same system or phenomenon can be represented in multiple different ways (graphically, tabulated, mathematically using different notational systems, etc.) that bring different facets or aspects of the system to the foreground. Likewise, we can, e.g. choose to use a certain synonym in a sentence or to structure a sentence in a certain way to emphasise a particular point in the present context.

The idea of semantics is in fact fairly close to the argument with which Redish and Kuo ( 2015 ) start their investigation: they argue that physicists use mathematics to represent physical systems and therefore load physical meaning onto mathematical symbols. They argue that in physics mathematical representations often bear different meanings than the same representations in the pure mathematics context. This is because, in forming a representation in a physics context we often blend physical and mathematical information. This usage of mathematics is what they intend to explicate in their modelling cycle, and it is strongly present also in our characterisation.

On a more general note, an entity in physics is made perceivable by expressing it in a mathematical form. Representations synthesise our understanding of the target, leaving out unnecessary information. The aim of a model can be, e.g. to represent experimental data in an ordered form or to write down a theoretical thought. It is possible to construct a theoretical representation of a phenomenon or system based only on the relevant physical background theory, or to construct an experimental representation where background theory and observations are in interaction with each other. This way mathematics allows constructing representations that can be further investigated and modified (2b).

Through representations, mathematics also provides a way to represent phenomena and systems in a shareable and reusable form; a mathematical representation is often easier to communicate to other people than, e.g. a qualitative description (2c). In that sense, mathematics acts as a common, universal language. For science in general, the communication function of mathematics is crucial.

3.2.1 Example: Matrix and Wave Mechanics

In the 1920s, the two most influential early formulations of quantum mechanics were developed. Born, Heisenberg, and Jordan refined Heisenberg’s initial work to a formulation commonly known as matrix mechanics, in which quantum mechanical quantities are expressed as matrices. According to Longair ( 2013 , p. 227), the aim in making the new formulation was “to rewrite all the equations of classical physics in matrix notation so that the key concept of non-commutativity [previously introduced in the work by Heisenberg] would automatically be incorporated in the new quantum mechanics”. Meanwhile, Schrödinger developed an alternative formulation of the theory, named wave mechanics. He started his work from the Hamilton–Jacobi differential equation, defining a new function, the “wave function”, and setting certain constraints for it. This way he was able to fulfil the pre-existing quantum conditions.

Even though the two formulations were later on found to be isomorphic in the context of the abstract Hilbert space theory of von Neumann, only Schrödinger’s wave mechanics stuck until today. The physicists at the time were reluctant to adopt matrix mechanics due to its unfamiliar mathematical machinery and cumbersome conceptual background. Ultimately, the matrix mechanical formalism was incorporated into the abstract Hilbert space formalism. We can regard the two early formalisms as representations that depicted the same theory but conveying their message differently. In the end, Schrödinger’s wave mechanics was more successful in expressing the theory in an understandable form.

The case of matrix and wave mechanics shows how the same theory can be mathematically represented in vastly different forms. It also illustrates the point that different mathematical representations may mediate differing aspects of the same theory. This contrasts with the idea that there are simply degrees of mathematisation (cf. the revised modelling cycle): mathematical and physical knowledge are intertwined in a more deep and nuanced way.

3.3 Reasoning and Modifications

A suitable representation can be further investigated through, e.g. mathematical reasoning and modifications. The chosen representation may either advance or hinder investigations and extraction of information, as different representations benefit different modes of reasoning. For example an algebraic representation allows mathematical manipulations more readily than a graphical representation. Thus, the possible reasoning and modification methods depend on the previous layers: the theory structure determines the available methods and rules for them, and the chosen representation limits the range of available methods.

In physics education, reasoning and modifications have a significant role. They are involved, for example in problem-solving situations in which mathematical representations are manipulated and analysed until a solution is found (Table 1 , role 3a). In physics research, mathematical representations are in a similar manner modified to test and expand the limits of physical models (3b). In these situations, the problem solver needs to apply their knowledge of the underlying rules and the physical context to the situation at hand. This enables using, e.g. logical reasoning in deducing new theoretical results (3c). Modifying and analysing mathematical representations may also aid in recognising new objects and regularities that would otherwise stay hidden (3d).

Following Easdown ( 2009 ), we separate two modes of reasoning: syntactic and semantic.

Syntactic reasoning and modifications utilise physical–mathematical rules in, e.g. carrying out calculations or modifying representations. This kind of use of mathematics requires little more than syntactic understanding; the semantic content of the studied representation does not enter the reasoning process. In other words, the reasoning process only applies rules stemming from the syntactic structure and does not utilise the understanding about the chosen representation. For example this may mean rote mathematical manipulations of equations in problem solving.

The separation between syntactic and semantic modes of reasoning seems to be rather widely acknowledged. In Easdown ( 2009 , p. 942), the same kind of (syntactic) reasoning is described as relying on “simple or naive, incremental rules, searching or pattern matching”. In physics education research, this kind of use of mathematics is often referred to as “plug and chug”, i.e. working through mathematics algorithmically. In the revised modelling cycle by Uhden et al. ( 2012 ), this step is detached from the physical–mathematical model, making a round to the area of “pure mathematics”. In their discussion, abilities of syntactic reasoning and modifications fall under the category of technical skills.

At the opposite end of the spectrum lies semantic reasoning. Here mathematics functions as a structure guiding and limiting thought. As the name suggests, semantic reasoning relies on understanding the semantics of the representation at hand, i.e. it requires understanding the content of the representation—or the physical meaning loaded onto mathematical symbols (Redish & Kuo, 2015 ). It heavily relies on the insight the reasoner has on the representation and on how the representation connects and maps onto the physical background theory of the studied system. One example of semantic reasoning is “opportunistic” blended reasoning (Hull et al., 2013 ), which relies on the physical interpretation of the used representation. Using this kind of reasoning, the reasoner may bypass the “doing mathematics” phase altogether during a problem-solving process.

Compared to the “plug and chug” aspect of syntactic reasoning, semantic reasoning may be conceptually harder to pin down. Easdown’s ( 2009 , p. 942) definition of semantic reasoning involves “solid intuition, insight or experience”, whereas de Regt ( 2017 , p. 105) describes an analogous reasoning mode as qualitative insight that “does not involve any calculations; it is based on general characteristics of the theoretical description of [the target]”. As de Regt also notes, at some point in a reasoning process, subsequent calculations are motivated and given direction by this qualitative reasoning.

As indicated earlier, the division between syntactic and semantic reasoning is not clear-cut. Instead, we can regard syntactic and semantic reasoning processes as opposite ends of the same spectrum, in the middle of which the reasoning modes gradually mix together. Most often in realistic situations, syntactic and semantic modes are both involved in the reasoning process and separating them from each other is artificial. These kinds of reasoning and modifications involve understanding both the syntactic and semantic content of the examined representation. The skills called structural in Uhden et al. ( 2012 ) probably lie somewhere in the middle of the spectrum.

3.3.1 Example: Wave Mechanics and Optics

An important part of Schrödinger’s justification for his wave mechanical formalism of quantum mechanics, as presented in the previous example, was an analogy with classical optics. His work was heavily influenced by de Broglie’s investigations, and originally he set out to find an equation describing de Broglie’s matter waves (Longair, 2013 ). As a result, he finally arrived at the non-relativistic Schrödinger equation.

According to Longair ( 2013 ), Schrödinger’s reasoning closely followed the classical path. Namely, classical ray optics that works well on small wave lengths compared to the system, e.g. a slit, breaks down when the wavelength and the scale of the system are of the same order, giving rise to the diffraction and interference phenomena. Analogously, he inferred, classical mechanics breaks down on small scales, and we can no longer neglect the wave properties of matter. The small enough scale to bring forth the quantum mechanical phenomena was reasoned to be of the same order as the de Broglie wavelength. This analogy with its mathematical machinery allowed Schrödinger to successfully justify his wave equation. The resulting formalism was not only successful in explaining previous observations but also “was based on the familiar apparatus of differential equations, akin to the classical mechanics of fluids and suggestive of an easily visualizable representation” (Jammer, 1989 , p. 270), which made it easier for physicists of the time to apply it.

Schrödinger’s use of analogical reasoning is an example of successfully blending syntactic and semantic reasoning. On one hand, he relied on the rules and limits set by the mathematical machinery, and on the other hand, he utilised his understanding of the content of the physical theories by letting the analogy to classical optics guide his work.

3.4 New Information and Explanations

A mathematical representation or its modifications can also give rise to new information. This gained information can be “new” in various ways: it can be a new scientific discovery or a new insight for a learner. In both cases, the representation or its modifications has yielded information that was previously more or less inaccessible.

Quite often in the context of physics education, solving typical end-of-chapter problems leads to clear numerical or analytic results. These solutions require interpretation in their context in order to give an answer to the original question. In this case, mathematics leads directly to new information. This way of ending up with a piece of information we call direct extraction of information. Similarly, scientific tests and analyses often lead to direct results (Table 1 , role 4a).

Sometimes, mathematical representations or their manipulations can uncover new structures that can lead to even unexpected explanations of phenomena. For example building a mathematical theory can result in theoretical predictions that foresee their observational confirmation (see the following example of the positron), in theoretical entities that have no direct physical interpretation (e.g. the wave function), or in a mathematical explanation of an observed phenomena that cannot be explained in qualitative terms (e.g. entanglement entropy). These kinds of objects could not have been constructed based solely on observations using the information available at the time. To emphasise the way these theoretical entities emerge from mathematics, we have chosen to call this pattern emergent extraction of information (4b–d).

As was the case with syntactic and semantic reasoning, the different modes of knowledge extraction are not clearly distinct. Again, they are rather the opposite ends of the same spectrum. We have, however, made the distinction to emphasise the emergent nature of many advancements of physical research: especially in these occasions mathematics has tremendously contributed to the scientific progress.

3.4.1 Example: Inventing the Positron

After the formation of matrix and wave mechanics, it was soon realised that achieving a relativistic quantum theory based on them was not a trivial task. In 1928, Dirac succeeded in generalising the Schrödinger equation into the relativistic Dirac equation. Even if it was not Dirac’s initial intention to incorporate spin into his theory or predict the existence of positrons, he ended up doing just that (Longair, 2013 ).

The first remarkable thing in the relativistic equation was that “in the relativistic formulation of quantum mechanics, there is necessarily a magnetic moment associated with the spin of the electron” and its magnitude coincided exactly with the observed value (Longair, 2013 , p. 338). Moreover, when solving for energies of the electron energy states from the Dirac equation, it emerges that there are also negative solutions. These negative energy solutions puzzled physicists of the time and it took some time for them to find an appropriate explanation for them. After a couple of years of pondering and proposing different explanations, Dirac came up with the proposal that the negative energies correspond to “a new kind of particle, unknown to experimental physics, having the same mass and opposite charge to an electron. We may call such a particle an anti-electron” (Dirac, 1931 ). Anti-electrons were later named positrons. Dirac’s invention of the positron illustrates how mathematical solutions can find interpretations in terms of novel physical conceptions.

4 Discussion

Especially in higher education, one aim of physics teaching is to guide students to understand physics as a field of science that has its distinctive characteristics. To convey this understanding, we need to do more than only train students to become fluent in modelling and problem solving. Even if those are important parts of the field, they offer too narrow a point of view. Consequently, physics education and educators should, both implicitly and explicitly, take into consideration the wider range of roles mathematics plays in physics. In a broad sense, this parallels or falls under the recommendation that educators should be aware of nature of science, if science education is to teach not only science but also “about” science. Furthermore, again analogously to common practices relating to NOS in education, the level of depth and abstraction in teaching about the entanglement of mathematics and physics can be adapted to students of various ages and knowledge levels. Here we have sought especially to aid educators in higher education to reflect on the issue, but hope to also contribute to, e.g. upper-secondary education in both physics and mathematics. For example the invention of new mathematical constructs and their subsequent adoption by natural scientists may be an unfamiliar yet intriguing idea for aspiring mathematicians.

In this article, we have presented a characterisation of how the roles of mathematics manifest on different levels in physics: at the level of theory structure, representations, mathematical reasoning, and new information and explanations. We echo arguments made by others in the same vein: e.g. Karam ( 2014 ) has discussed similar roles in their research. However, we wish to add to this discussion by introducing with our characterisation a more organised view of the roles mathematics plays in physics. In addition, we wish to diversify discourses of mathematics in physics and physics teaching by stressing issues such as the extraction of information through mathematics.

Comparing the new characterisation with the previous models, e.g. the modelling cycles of Redish and Kuo ( 2015 ) and Uhden et al. ( 2012 ), it is apparent that they are meant to answer somewhat different questions: the modelling cycles focus on modelling and problem-solving actions and skills, whereas our characterisation aims to provide a more general description of the roles mathematics takes in physics. Most notably, while the modelling cycle of Redish and Kuo ( 2015 ) highlights building models and performing mathematical manipulations, it differs from our characterisation as it does not bring up the significance of theory structure, differentiate between modes of reasoning, or emphasise the extraction of mathematical knowledge. In part, Uhden et al. ( 2012 ) address some of these shortages and further emphasise the reasoning processes in terms of separate structural and technical use of mathematics. However, neither do they include theory structure explicitly in their model nor cover different ways mathematics leads to new knowledge. Pointing the explicit differences between the existing models and our characterisation, however, is not to say that these parts are necessarily completely missing in the previous models, as they can be seen as included implicitly. For example the contribution of theory structure is inevitably present in parts of the models as an element that guides modelling and interpretation processes or mathematical manipulations. However, our chosen epistemological perspective, i.e. the premise that mathematics serves as a constitutive structure in physics analogous to language, brings these different roles to the foreground.

Therefore, our characterisation also yields implications for physics education research and physics instruction, as it can help researchers as well as instructors avoid what Karam ( 2014 , p. 1) calls “an artificial separation between the mathematical and the conceptual aspects of physical theories”. Assuming our proposed standpoint, it becomes unreasonable to separate these aspects. For physics education research, in which conceptual and mathematical (or procedural) learning are often seen as separate, this would mean shifting the focus to a more blended view, in which physics learning is regarded as profoundly dependent on both the aspects. This would also force us to rethink physics instruction, taking this entanglement into account and making it explicit in teaching.

Besides emphasising the intertwining of conceptual and mathematical aspects, our characterisation provides a holistic view of physical theories: physical theories are not just unrelated pieces, but parts of a larger structure held together, organised and expressed by mathematics. Using this framework in physics learning would, we argue, likely lead to deeper and more coherent understanding, as this framework provides a natural basis on which to build new knowledge. This is also something to be explicated in physics instruction.

These lines of argument gain further importance in the case of more advanced physics theories because conceptual physical and formal mathematical aspects are so inextricably merged in them. How does one explain concepts such as a spacetime metric or the state of a quantum system without mathematics? We argue that making the entanglement of these aspects explicit in the instruction of these theories would be especially beneficial for student learning. Here, both general and highly domain-specific examinations of the interplay of mathematics and physics may be needed, with content domains spanning various stages of science learning; after all, the guidelines given here are intended primarily for students on and around the undergraduate level. At the moment, however, physics education research in the field of advanced physical theories is relatively scarce, and in order to research learning in these fields and to make more robust arguments in relation to learning advanced physics, we must adopt a wider view on the roles mathematics plays in physics and physics learning. Focusing only on the modelling and problem-solving perspectives (emphasising the role of mathematics as a tool) may even hinder the advancement of educational research into the more advanced physics theories, while potentially oversimplifying and misrepresenting the interplay of the disciplines.

5 Conclusions

The aim of physics as a field of science is to gain understanding of physical phenomena. To accumulate this collective understanding, the scientific community needs to construct new explanations that can be communicated and organised—and in these construction, communication and organisation processes mathematics plays an important role. Mathematics does not only present physical information, but also shapes our beliefs about physical phenomena. Because mathematics is so tightly knit into the formation of physical knowledge, its role is inextricably intertwined also with physics learning. As we have demonstrated, this dynamic is more complex than typical problem-solving or modelling-centred perspectives account for, and thus there is a need for more comprehensive characterisations. Here we have illustrated the wider array of the roles of mathematics in physics, but further work is needed to examine how to incorporate representation of and discourse on these roles in how we teach the nature of science and the philosophy of physics.

In order to make this constructive role of mathematics more explicit, we have presented a new characterisation of how the roles of mathematics appear on different levels of physics. It also brings forward the parts that the previous problem-solving or modelling-oriented models have neglected, namely, theory structure and the emergence of new knowledge. Moreover, it emphasises the gradual mixing of syntactic and semantic reasoning. These are key features in the development of understanding in more advanced physics theories: the more advanced a theory is, the more entangled the syntactic and semantic modes are in reasoning—as we have sought to illustrate through examples from the history of modern physics. Likewise, the emergent nature of many entities is more prominent in these topics. In order to gain ground in studying the learning of these more advanced topics, such as quantum physics and relativity, we have to take these perspectives into account. In our view, to ensure that pedagogies comprehensively address these “entangled” bodies of knowledge, educators should be aware of a fuller range of roles of mathematics in physics.

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Acknowledgements

We are grateful for helpful comments and discussions with Ricardo Karam, Inkeri Kontro, Tommi Kokkonen, and Ismo T. Koponen.

This work has been supported in part by a grant from Magnus Ehrnrooth Foundation. Open Access funding provided by University of Helsinki including Helsinki University Central Hospital.

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Palmgren, E., Rasa, T. Modelling Roles of Mathematics in Physics. Sci & Educ 33 , 365–382 (2024). https://doi.org/10.1007/s11191-022-00393-5

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